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Frustration and thermalization in an artificial magnetic quasicrystal

Nature Physics volume 14, pages 309–314 (2018)Cite this article

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Abstract

Artificial frustrated systems offer a playground to study the emergent properties of interacting systems. Most work to date has been on spatially periodic systems, known as artificial spin ices when the interacting elements are magnetic. Here we have studied artificial magnetic quasicrystals based on quasiperiodic Penrose tiling patterns of interacting nanomagnets. We construct a low-energy configuration from a step-by-step approach that we propose as a ground state. Topologically induced emergent frustration means that this configuration cannot be constructed from vertices in their ground states. It has two parts, a quasi-one-dimensional ‘skeleton’ that spans the entire pattern and is capable of long-range order, surrounding ‘flippable’ clusters of macrospins that lead to macroscopic degeneracy. Magnetic force microscopy imaging of Penrose tiling arrays revealed superdomains that are larger for more strongly coupled arrays, especially after annealing the array above its blocking temperature.

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Fig. 1: Penrose tiling pattern and the seven types of vertex found within it.
Fig. 2: The lowest energy configuration of each vertex and a low-energy state of the entire pattern.
Fig. 3: Ordered regions in the skeleton of an artificial magnetic quasicrystal.
Fig. 4: Populations of the energy levels of the seven vertex types.
Fig. 5: The blocking temperature and micromagnetic simulation results of the energy population distribution for type VII vertices.

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References

  1. Bernal, J. D. & Fowler, R. H. A theory of water and ionic solution, with particular reference to hydrogen and hydroxyl ions. J. Chem. Phys. 1, 515–548 (1933).

    Article  ADS  Google Scholar 

  2. Harris, M. J., Bramwell, S. T., McMorrow, D. F., Zeiske, T. & Godfrey, K. W. Geometrical frustration in the ferromagnetic pyrochlore Ho2Ti2O7. Phys. Rev. Lett. 79, 2554–2557 (1997).

    Article  ADS  Google Scholar 

  3. Wang, R. F. et al. Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands. Nature 439, 303–306 (2006).

    Article  ADS  Google Scholar 

  4. Han, Y. et al. Geometric frustration in buckled colloidal monolayers. Nature 456, 898–903 (2008).

    Article  ADS  Google Scholar 

  5. Latimer, M. L., Berdiyorov, G. R., Xiao, Z. L., Peeters, F. M. & Kwok, W. K. Realization of artificial ice systems for magnetic vortices in a superconducting MoGe thin film with patterned nanostructures. Phys. Rev. Lett. 111, 067001 (2013).

    Article  ADS  Google Scholar 

  6. Ortiz-Ambriz, A. & Tierno, P. Engineering of frustration in colloidal artificial ices realized on microfeatured grooved lattices. Nat. Commun. 7, 10575 (2016).

    Article  ADS  Google Scholar 

  7. Nisoli, C., Moessner, R. & Schiffer, P. Colloquium: Artificial spin ice: Designing and imaging magnetic frustration. Rev. Mod. Phys. 85, 1473 (2013).

    Article  ADS  Google Scholar 

  8. Morgan, J., Stein, A., Langridge, S. & Marrows, C. H. Thermal ground state ordering and elementary excitations in artificial magnetic square ice. Nat. Phys. 7, 75–79 (2011).

    Article  Google Scholar 

  9. Morgan, J. P. et al. Real and effective thermal equilibrium in artificial square spin ices. Phys. Rev. B 87, 024405 (2013).

    Article  ADS  Google Scholar 

  10. Porro, J., Pinto, A., Berger, A. & Vavassori, P. Exploring thermally induced states in square artificial spin-ice arrays. New J. Phys. 15, 055012 (2013).

    Article  ADS  Google Scholar 

  11. Farhan, A. et al. Direct observation of thermal relaxation in artificial spin ice. Phys. Rev. Lett. 111, 057204 (2013).

    Article  ADS  Google Scholar 

  12. Kapaklis, V. et al. Thermal fluctuations in artificial spin ice. Nat. Nanotech. 9, 514–519 (2014).

    Article  ADS  Google Scholar 

  13. Perrin, Y., Canals, B. & Rougemaille, N. Extensive degeneracy, Coulomb phase and magnetic monopoles in artificial square ice. Nature 540, 410–413 (2016).

    Article  ADS  Google Scholar 

  14. Tanaka, M., Saitoh, E., Miyajima, H., Yamaoka, T. & Iye, Y. Domain structure and magnetic ice-order in NiFe nano-network with honeycomb structure. J. Appl. Phys. 97, 10J710 (2005).

    Article  Google Scholar 

  15. Qi, Y., Brintlinger, T. & Cumings, J. Direct observation of the ice rule in an artificial kagome spin ice. Phys. Rev. B 77, 094418 (2008).

    Article  ADS  Google Scholar 

  16. Ladak, S., Read, D. E., Perkins, G. K., Cohen, L. F. & Branford, W. R. Direct observation of magnetic monopole defects in an artificial spin-ice system. Nat. Phys. 6, 359–363 (2010).

    Article  Google Scholar 

  17. Mengotti, E. et al. Real-space observation of emergent magnetic monopoles and associated Dirac strings in artificial kagome spin ice. Nat. Phys. 7, 68–74 (2011).

    Article  Google Scholar 

  18. Rougemaille, N. et al. Artificial kagome arrays of nanomagnets: A frozen dipolar spin ice. Phys. Rev. Lett. 106, 057209 (2011).

    Article  ADS  Google Scholar 

  19. Zhang, S. et al. Crystallites of magnetic charges in artificial spin ice. Nature 500, 553–557 (2013).

    Article  ADS  Google Scholar 

  20. Möller, G. & Moessner, R. Magnetic multipole analysis of kagome and artificial ice dipolar arrays. Phys. Rev. B 80, 140409 (2009).

    Article  Google Scholar 

  21. Chern, G.-W., Mellado, P. & Tchernyshyov, O. Two-stage ordering of spins in dipolar spin ice on the kagome lattice. Phys. Rev. Lett. 106, 207202 (2011).

    Article  ADS  Google Scholar 

  22. Anghinolfi, L. et al. Thermodynamic phase transitions in a frustrated magnetic metamaterial. Nat. Commun. 6, 8278 (2015).

    Article  Google Scholar 

  23. Nisoli, C. et al. Ground state lost but degeneracy found: the effective thermodynamics of ‘artificial spin ice’. Phys. Rev. Lett. 98, 217103 (2007).

    Article  Google Scholar 

  24. Budrikis, Z. et al. Disorder strength and field-driven ground state domain formation in artificial spin ice: experiment, simulation and theory. Phys. Rev. Lett. 109, 037203 (2011).

    Article  ADS  Google Scholar 

  25. Kapaklis, V. et al. Melting artificial spin ice. New J. Phys. 14, 035009 (2012).

    Article  ADS  Google Scholar 

  26. Farhan, A. et al. Exploring hyper-cubic energy landscapes in thermally active finite spin-ice systems. Nat. Phys. 9, 375–382 (2013).

    Article  Google Scholar 

  27. Chioar, I. A. et al. Kinetic pathways to the magnetic charge crystal in artificial dipolar spin ice. Phys. Rev. B 90, 220407 (2014).

    Article  ADS  Google Scholar 

  28. Drisko, J., Daunheimer, S. & Cumings, J. FePd3 as a material for studying thermally active artificial spin ice systems. Phys. Rev. B 91, 224406 (2015).

    Article  ADS  Google Scholar 

  29. Morley, S. A. et al. Vogel–Fulcher–Tammann freezing of a thermally fluctuating artificial spin ice probed by X-ray photon correlation spectroscopy. Phys. Rev. B 95, 104422 (2017).

    Article  ADS  Google Scholar 

  30. Morrison, M. J., Nelson, T. R. & Nisoli, C. Unhappy vertices in artificial spin ice: new degeneracies from vertex frustration. New J. Phys. 15, 045009 (2013).

    Article  ADS  Google Scholar 

  31. Gilbert, I. et al. Emergent ice rule and magnetic charge screening from vertex frustration in artificial spin ice. Nat. Phys. 10, 671–676 (2014).

    Article  Google Scholar 

  32. Gilbert, I. et al. Emergent reduced dimensionality by vertex frustration in artificial spin ice. Nat. Phys. 12, 162–165 (2016).

    Article  Google Scholar 

  33. Wang, Y.-L. et al. Rewritable artificial magnetic charge ice. Science 352, 962–966 (2016).

    Article  ADS  Google Scholar 

  34. Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953 (1984).

    Article  ADS  Google Scholar 

  35. Janot, C. Quasicrystals: A Primer (Clarendon, Oxford, 1992).

  36. Charrier, B. & Schmitt, D. Dynamical and irreversible magnetic effects in Ir8Mg42Zn50 quasicrystals (R=Tb, Dy). J. Magn. Magn. Mater. 189, 165–172 (1998).

    Article  ADS  Google Scholar 

  37. Islam, Z. et al. Reinvestigation of long-range magnetic ordering in icosahedral Tb-Mg-Zn. Phys. Rev. B 57, 11047–11050 (1998).

    Article  ADS  Google Scholar 

  38. Sato, T. Short-range order and spin-glass-like freezing in a-mg-r (a=Zn or Cd; r=rare-earth elements) magnetic quasicrystals. Acta Cryst. A A61, 39–50 (2005).

    Article  Google Scholar 

  39. Bhat, V. S. et al. Controlled magnetic reversal in permalloy films patterned into artificial quasicrystals. Phys. Rev. Lett. 111, 077201 (2013).

    Article  ADS  Google Scholar 

  40. Farmer, B. et al. Direct imaging of coexisting ordered and frustrated sublattices in artificial ferromagnetic quasicrystals. Phys. Rev. B 93, 134428 (2016).

    Article  ADS  Google Scholar 

  41. Brajuskovic, V., Barrows, F., Phatak, C. & Petford-Long, A. K. Real-space observation of magnetic excitations and avalanche behavior in artificial quasicrystal lattices. Sci. Rep. 6, 34384 (2016).

    Article  ADS  Google Scholar 

  42. Vedmedenko, E. Y., Oepen, H. P. & Kirschner, J. Decagonal quasiferromagnetic microstructure on the Penrose tiling. Phys. Rev. Lett. 90, 137203 (2003).

    Article  ADS  Google Scholar 

  43. Nisoli, C. On thermalization of magnetic nano-arrays at fabrication. New J. Phys. 14, 035017 (2012).

    Article  ADS  Google Scholar 

  44. Wang, R. F. et al. Demagnetization protocols for frustrated interacting nanomagnet arrays. J. Appl. Phys. 101, 09J104 (2007).

    Article  Google Scholar 

  45. Donahue, M. J. & Porter, D. G. OOMMF User’s Guide, Version 1.0. Interagency Report NISTIR 6376 (National Institute of Standards and Technology, Gaithersburg, MD, 1999); http://math/nist.gov/oommf

  46. Gummelt, P. Penrose tilings as coverings of congruent decagons. Geometriae Dedicata 62, 1–17 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  47. Vedmedenko, E. Y., Grimm, U. & Wiesendanger, R. Noncollinear magnetic order in quasicrystals. Phys. Rev. Lett. 93, 076407 (2004).

    Article  ADS  Google Scholar 

  48. Ke, X. et al. Energy minimization and ac demagnetization in a nanomagnet array. Phys. Rev. Lett. 101, 037205 (2008).

    Article  ADS  Google Scholar 

  49. Osborn, J. A. Demagnetizing factors of the general ellipsoid. Phys. Rev. 67, 351–357 (1945).

    Article  ADS  Google Scholar 

  50. Morgan, J. P., Bellew, A., Stein, A., Langridge, S. & Marrows, C. H. Linear field demagnetization of artificial magnetic square ice. Front. Phys. 1, 28 (2013).

    Article  Google Scholar 

Download references

Acknowledgements

This research used resources of the Center for Functional Nanomaterials, which is a US DOE Office of Science Facility, at Brookhaven National Laboratory under Contract No. DE-SC0012704, and was also supported by the EPSRC (grant EP/L00285X/1). Z.B. was supported by the ERC Advanced Grant no. 291002 SIZEFFECTS.

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Authors and Affiliations

  1. School of Physics & Astronomy, University of Leeds, Leeds, UK

    Dong Shi, Sophie A. Morley, Peter D. Olmsted, Gavin Burnell & Christopher H. Marrows

  2. Institute for Scientific Interchange Foundation, Torino, Italy

    Zoe Budrikis

  3. Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, NY, USA

    Aaron Stein

  4. Department of Physics, Institute for Soft Matter Synthesis and Metrology, Georgetown University, Washington DC, USA

    Peter D. Olmsted

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  1. Dong Shi
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Contributions

C.H.M. conceived the project. A.S. fabricated the samples, using Penrose array designs made by G.B. and D.S. D.S. performed the MFM imaging and the a.c.-demagnetization and thermal annealling protocols, analysed the data (using software co-designed with G.B.) and theoretically derived the expected ground state. Z.B. performed the Monte Carlo simulations in consultation with D.S. S.A.M. performed the SQUID magnetometry measurements. C.H.M., G.B. and P.D.O. supervised the work. All authors discussed the data and commented on the manuscript.

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Correspondence to Christopher H. Marrows.

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Supplementary notes 1-7, Supplementary Figures 1-14, and Supplementary References 1-15

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Shi, D., Budrikis, Z., Stein, A. et al. Frustration and thermalization in an artificial magnetic quasicrystal. Nature Phys 14, 309–314 (2018). https://doi.org/10.1038/s41567-017-0009-4

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